You'll need some Slime Keys to open Slime Gates.

You can obtain these by feeding Gordo Slimes (the large slimes encountered around the map) until they explode. You need to feed them 50 items of food (fruit, meat or vegetables), or 25 of their favourite food. The Pink Gordo only needs 30 items of food and doesn't have a favourite. 

Please note that not all Gordo Slimes give keys. You can only get a key from the following Gordo Slimes:

• Pink Gordo #1
• Pink Gordo #2
• Phosphor Gordo
• Honey Gordo
• Hunter Gordo
• Crystal Gordo
• Rad Gordo
• Boom Gordo
• Quantum Gordo
• Dervish Gordo
• Mosaic Gordo